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# Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.$y = 3 - 2 \cos x$

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our coolest a graph dysfunction, but we're going to start with the graph of one of the standard functions Y equals co Sign X and then figure out the transformations that took place. So the coastline graph has a period of two. Pi goes up to one and down to negative one and looks roughly like this. It goes on in both directions, but I'm just showing part of it. Okay, now what happens if we multiply it by two When we multiply it by two. That's a vertical stretch, so now it's going to be twice as tall. So now it's going to go up to a height of two and down to a height of negative, too. We say the amplitude changed. Okay, so we can go ahead and sketch that twice is tall. All right, now what happens if we multiply that by negative one? Well, that's going to reflect it across the X axis. Flip it upside down, you could say. So we'll take what we just drew and make that move still going up to a height of one at a height of two down to a height of negative, too, but upside down to direct sketch. Okay? Finally, the graph that we've been waiting for Why equals three minus to co sign X. All we have to do now is at the three and adding the three is going to shift the whole graph of three. So now instead of the top point being at a height of two, the top point is going to be at her five a height of five. And instead of the bottom point being at a height of negative two, it's going to be at a height of one. So let's make room for that. So the middle line is going to be shifted up. Three. I'll just draw in that middle line there to give us some reference. Okay, so we have this upside down graph going down to a height of one and up to a height of five. There we go.

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